# A comparison of Futhark and Dex

Dex is a functional array programming language developed by a team of researchers at Google. I recently re-read their paper, which got me excited enough to want to take a closer look. Dex and Futhark are more or less aimed at the same kinds of problems, so my interpretation of Dex is rooted in how it differs from Futhark. In this post I will describe some of the interesting differences based on translating five Dex example programs to Futhark. I’m not a Dex expert, so maybe I’ve missed a thing here or there.

Futhark wasn’t originally designed to be a user-facing programming language. We were doing research in compiler optimisations for parallel computers, and the language was just a crude little thing so we could write programs for our optimiser to work on. Over time the language grew and eventually became fairly pleasant to use (full story here), but it was still never designed as a cohesive or novel approach to array programming. That also means it’s fairly conventional or even old-fashioned, as functional languages go. In contrast, Dex’s authors had more imagination and designed their language from the start with novel ideas, chief of which is to consider *index sets as types*. To illustrate the idea, here is how to compute all-pairs L₁ distances in Dex:

```
pairwiseL1 :: n=>d=>Real -> n=>n=>Real
pairwiseL1 x = for i j.sum (for k. abs (x.i.k - x.j.k))
```

The `n=>d=>Real`

is the type of an `n`

by `d`

array of `Real`

s. Dex leans heavily on an analogy between arrays and functions, as arrays can be seen as merely functions from indexes to values. In Futhark, we’d write this type as `[n][d]Real`

. Note that in Dex, `n`

and `d`

are completely abstract type parameters, while in Futhark they are term-level variables.

The real advantage of Dex’s approach is that it permits a very lightweight notation for index spaces. For example, `for i j.e`

produces a two-dimensional array where each element is given by the expression `e`

, and the type checker figures out the span of `i`

and `j`

based on the context. For example, in `for k`

, Dex figures out that `k`

must be part of the index set `d`

, because it is used to index the innermost dimension of `x`

. Pretty cool!

A naive translation to Futhark would be this:

```
let pairwiseL1 [n][d] (x: [n][d]f64) =
2d n n (\i j -> f64.sum (tabulate d (\k -> x[i,k] - x[j,k]))) tabulate_
```

Note that the tabulation functions require explicit size-passing, and that the indexes are just integers - the type checker will not help us if we accidentally use the `k`

along the wrong dimension.

Of course, the above is not how you’d actually write this program in Futhark. Instead you’d first define a function for computing the L₁ distance:

```
let L1 [n] (xs: [n]f64) (ys: [n]f64) : f64 =
map2 (-) xs ys |> map f64.abs |> f64.sum
```

And then you’d apply it to all the pairs:

```
let pairwiseL1 [n][m] (xss: [n][m]f64) : [n][n]f64 =
map (\a -> map (\b -> L1 a b) xss) xss
```

I think this program illustrates the main difference in philosophy between Dex and Futhark. While Dex uses dependent types to secure an index-based notation, Futhark instead encourages index-free programming. I suspect the two approaches are fundamentally equivalent, but it’s an interesting contrast that I think is due to the two language’s different backgrounds. Dex is specifically designed to implement scientific code and formulae, which is traditionally very index-oriented. Futhark is more about supporting a “traditional” combinator-based functional programming style, but just making it run much faster. You could view Futhark as a data-parallel ML, while Dex is higher-order dependently typed Einstein summation.

I also suspect this focus on indexes is because the Dex authors have a background of being frustrated with NumPy-style programming, where the absence of efficient indexing can be quite restrictive. They even even use this NumPy implementation of L₁ distances as motivation in their paper:

```
def pairwiseL1(x):
return sum(abs(x.T - x[..., newaxis]), axis=1)
```

I certainly agree that this is hard to read.

## The good ones

Porting a two-line Dex program to Futhark is enough to wax philosophically for a paragraph or two, but it’s still a pretty shallow comparison. Therefore, I also ported five of the Dex example programs, plus whatever of the Dex prelude I needed along the way. I’m not going to claim that I ported the five most difficult programs, but at least one of them was quite complicated. The Futhark programs total about 450 lines of code (excluding comments and blanks).

My general impression is that when it comes to expressing parallelism, Dex and Futhark are about equivalent. Dex’s index notation is more concise, but I personally find it slightly easier to understand and decompose Futhark expressions. As an example, this Dex function computes the covariance of a matrix:

```
def covariance (n:Type) ?-> (d:Type) ?->
(xs:n=>d=>Float) : (d=>d=>Float) =
xsMean : d=>Float = (for i. sum for j. xs.j.i) / IToF (size n)
xsCov : d=>d=>Float = (for i i'. sum for j.
(xs.j.i' - xsMean.i') *
(xs.j.i - xsMean.i ) ) / IToF (size n - 1)
xsCov
```

In Futhark we write it as:

```
let covariance0 [n] (xs:[n]f64) (xsm:f64) (ys:[n]f64) (ysm:f64) =
-1)
f64.sum (map2 (\x y -> (x-xsm) * (y-ysm)) xs ys) / f64.i64 (n
let covariance [n][d] (xs:[n][d]f64) =
let xsT = transpose xs
let means = map mean xsT
in map2 (\a a_mean ->
map2 (\b b_mean -> covariance0 a a_mean b b_mean)
xsT means) xsT means
```

It’s certainly more verbose, but I had to read the Dex function carefully to understand what the indexes implied, while I have a much easier time understanding the structure of the computation from the Futhark formulation. Of course, I also have years of experience with Futhark, compared to just days with Dex.

Most of the translations were pretty simple, for example the Mandelbrot set, Monte Carlo pi, and Brownian motion programs. One difference that made me feel *major* jealousy is that the `dex script`

command is also able to generate pleasant reports containing both the code and visualisations and plots of various values. We definitely need a tool like this for Futhark!

The Sierpinski triangle program has a fun little detail in Dex, which is that the `randIdx`

function uses the Dex type system to determine the range of the index being produced. While the `randIdx`

function itself can still be wrong, this makes it hard to *use* it incorrectly. The Futhark translation of `randIdx`

asks the user to pass in a range explicitly, and also returns just an integer.

## The bad one

The largest ported example by far is a ray tracer. It uses ray marching with signed distance functions to describe objects. The Dex program rather casually uses the `grad`

operator to apply automatic differentiation (AD) to compute surface normals from the distance function. This is a really elegant technique, but Futhark does not (yet!) have a `grad`

operator. In Futhark, the sensible thing to do is to hard-code the gradient functions for the three different kinds of objects, so of course I instead used forward-mode AD with dual numbers implemented via the Futhark module system. The resulting code finally convinced me that built-in AD is a necessity for a modern numerical languages. I was on the fence before, since I worry that doing it well will be invasive in both the language and compiler, but I never want to write this kind of boilerplate again.

The rest of the ray tracer was fairly straightforward to implement. Dex uses its effect system to implement the loop where the lights in the scene apply their contributions to a given point, which I wrote in Futhark as basically a fold. In fact, I didn’t yet find a Dex example where the effect system was more than a small notational convenience. I’m sure there’s one, though! Effect systems are not things you just add on a lark.

There was one part that confused me initially, but which makes perfect sense in retrospect. The ray tracer normalises the intensity of all pixels (triples of floats) based on the average intensity (unusual I think, but fine). In Dex this is done like this:

`image / mean (for (i,j,k). image.i.j.k)`

When I first read this, I couldn’t figure out whether it was normalising *per channel*. I always get a bit wary when overloaded operators like that `/`

are involved. Of course, that `for`

-expression is over a *single* index that just happens to be a triple, and the components of which are then used to index the three-dimensional `image`

array. It’s really just flattening the array, and the type checker makes the individual `i`

, `j`

and `k`

s take on the appropriate value.

## Conclusions

With respect to expressing parallelism, Dex and Futhark seem equivalent in expressive power, but Dex has the edge in concision. I’d be curious about going the other way, and porting some of the original Futhark benchmark programs *to* Dex, like local volumetric calibration.

Dex has several small conveniences over Futhark: while the effect system didn’t matter much for the examples I looked at, Dex’s type classes and broadcasting operators did help a bit with making things more concise.

If you need AD, then Dex is miles ahead of Futhark. While I managed to implement the surface normals in the ray tracer, I gave up on porting mcmc.dx because it contains a higher-order function that applies the `grad`

operator to a functional argument. This would have to be implemented with a higher order parametric module (which I wrote were useless not long ago), but I just didn’t have the heart for it. I’ll keep this as a usage case for when we implement AD properly.

I didn’t look much at performance, since Dex is sparsely documented and the benchmarking tools seem to be mostly for internal use. I performed a rough timing of sequential execution of the ray tracer, where the Futhark and Dex versions are about equally fast. Dex also has multi-threaded and CUDA backends, but I did not try them.

Speaking of sparse documentation, Dex is still young and appears to be changing frequently. My understanding is based on the paper, reading the example programs, and skimming some of the implementation. I may have missed important details, and this post may even be outdated by the time you read it.

### Inspiration for Futhark

I don’t think it would take that much effort to let Futhark be more implicit with respect to sizes. An easy start would be to permit return-size polymorphism, which would let us write a `tabulate`

function with this type:

`val tabulate [n] 'a : (i64 -> a) -> [n]a`

Currently the Futhark type checker forbids size parameters that are used only in negative position. This restriction is mostly because I implemented the size type system on my own, and since I had little experience with implementing dependent type systems, I was worried about inadvertently admitting unsound constructs (and more pragmatically, I was worried about bugs in the implementation). I locked it down more than might strictly be necessary. If we want to support this kind of result size inference, then we do need to figure out what to do with expressions like:

` zip (tabulate f) (filter p xs)`

Here the size of the `tabulate`

must be the size of the array returned by `filter`

, which is existential. As far as I can figure based on the paper, Dex wouldn’t allow an expression like the above, as it handles existentials in a conventional explicit manner:

`filter :: (a -> Bool) -> m=>a -> E n. n=>a`

Presumably it’s up to the user to do the unpacking of the existential context as needed. In Futhark I wanted to preserve “direct style” programming, so you could write expressions like `map f (filter p xs)`

where the compiler implicitly unpacks the existential context for you, but maybe the cost is too great.

I think *some degree* of Dex’s type-safe indexing can almost already be implemented in Futhark, especially if we loosen the above restriction that size parameters must be used in parameters. We already have an example where “phantom sizes” are used to implement triangular arrays. Maybe Futhark should go more in the direction of dependent types? It’s unfortunate that we’re a small team, as it also takes a lot of time to write papers about making the language go wrooooom…